Isolated point

A point of a subset having a neighborhood that contains no other point of that subset.

Version

v2 · 2026-09-19 · History

Domain-specific #

5121

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

General Topology → Mathematics

Core Idea

For A in a topological space X, x∈A is isolated when some open neighborhood U of x satisfies U∩A={x}, equivalently {x} is open in the subspace topology on A. The ambient topology supplies neighborhoods; restricting to the subset makes the singleton locally visible and prevents sequences of distinct A-points from converging to x under suitable countability conditions. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

The Lonely Dot

Imagine red dots scattered on a page. A red dot is isolated if you can draw a little circle around it that has no other red dots inside. Even if the page is full of other stuff, no other red dot is right next to it.

Point With No Neighbors

Pick a group of points, called a set, inside a bigger space like a line or a sheet of paper. A point in the group is isolated if you can find a small region around it that contains no other points from the group. For example, in the set made of all numbers between 0 and 1 plus the number 5, the number 5 is isolated. The numbers between 0 and 1 are not isolated, because any region around one of them always catches other members. Whether a point counts as isolated depends on which group you are looking at.

Non-Limit Point of a Set

Take a set A inside a bigger space X where we know what 'nearby' means, for example the real number line. A point x in A is isolated if some open neighborhood of x contains no point of A except x itself. For example, in the set {0} together with all numbers 1/n, each 1/n is isolated, but 0 is not, because points 1/n get arbitrarily close to it. A point being isolated depends on which set A you are looking at: the same point might be isolated in one set and not in another. Isolated points are the opposite of limit points of A.

 

Let X be a topological space and A ⊆ X. A point x ∈ A is isolated in A if there is an open set U in X with U ∩ A = {x}. Equivalently, the singleton {x} is open in the subspace topology on A. The ambient topology supplies the neighborhoods, while the subset determines which other points must be excluded. Consequently, under suitable countability conditions, no sequence of distinct points of A converges to x. The notion requires explicit specification of the space, its topology or metric, the subset, the point's membership, and the isolating neighborhood. For example, every point of ℤ is isolated in ℝ's subspace topology, while 0 is not isolated in {0} ∪ {1/n}. A set in which every point is isolated carries the discrete subspace topology.

Scope of Application

Isolated point belongs to general topology and is useful where the analyst can specify the typed general topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate ambient space, subset, point membership, topology or metric, and isolating neighborhood are explicit and exclude every other subset point. The scope is broad within that domain but bounded by the need for ambient space, subset, point membership, topology or metric, and isolating neighborhood are explicit and exclude every other subset point. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making ambient space, subset, point membership, topology or metric, and isolating neighborhood are explicit and exclude every other subset point the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Isolated point can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Isolated point. Isolated point compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed general topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express ambient space, subset, point membership, topology or metric, and isolating neighborhood are explicit and exclude every other subset point independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of general topology because they reuse the typed general topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, The ambient topology supplies neighborhoods; restricting to the subset makes the singleton locally visible and prevents sequences of distinct A-points from converging to x under suitable countability conditions., and type the carrier, state every parameter and convention in the definition, test that ambient space, subset, point membership, topology or metric, and isolating neighborhood are explicit and exclude every other subset point, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Isolated point Domain-specific

Parents (1) — outgoing typed relations

• Isolated point is a kind of Discreteness Prime

  What makes it its own entry: the domain-specific identity determined by ambient space, subset, point membership, topology or metric, and isolating neighborhood are explicit and exclude every other subset point.

Hierarchy paths (2) — routes to 2 parentless roots

• Isolated point → is a kind of Discreteness → is a kind of Boundary

Show alternative path (1)

Neighborhood in Abstraction Space

Isolated point sits in a crowded region of the domain-specific corpus (1^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Point-Set Topology Structures (51 abstractions)

Nearest neighbors

• Adherent point — 0.96
• Regular space — 0.95
• First-countable space — 0.95
• Discrete space — 0.94
• Door space — 0.93

Computed from structural-signature embeddings · 2026-10-25